A frequency-domain analytical solution of Maxwell's equations for layered anisotropic media

Reconstruction of pressure and shear velocities and boundaries of thin layers in a thinly stratified stack

“…Before describing the numerical experiment, we have to indicate how we solve the direct problems (5)-(8), (11)-(14), and (23)- (26), find the gradients of the residual functionals (10) and (15) with variations of the velocities v p and v s , and calculate the derivative (37).…”

“…This approach is based on the results of [20][21][22][23][24][25][26][27][28]. Using this method for numerical solving the above-posed problems, we can obtain analytical expressions for their solutions, and the search for these solutions reduces to recurrent recalculation from one layer to another; moreover, these analytical expressions can be presented in such a form that the rounding error is not accumulated when passing from one layer to another.…”

“…Using this method for numerical solving the above-posed problems, we can obtain analytical expressions for their solutions, and the search for these solutions reduces to recurrent recalculation from one layer to another; moreover, these analytical expressions can be presented in such a form that the rounding error is not accumulated when passing from one layer to another. To calculate derivative (37), we have to solve the direct problem (23)- (26) and to known the residual. The gradient for the residual functional (15) with variations of the velocity v p can be obtained in an analytical form.…”

“…The least accuracy is observed for the boundary location in the case with a small difference in the velocities. As is seen from the formulation of problem (23)- (26), the derivative of the residual functional with respect to the coordinate of the boundary point can vanish not only at the minimum point, but also in the case where the boundary is absent, i.e., if the following equalities are satisfied at a certain point z i :…”

Simultaneous reconstruction of permittivity and conductivity

“…This method was successfully used, for example, in [21][22][23][24]. For solving differential equation (1.1) we introduce a function s as follows…”